Question

a scientist records the number of cells in a culture. the function ( f(x)=1,600(1.35)^{x} ) describes ( f ), the number of cells in the culture after ( x ) hours. which statement best describes one value in this function? the number of cells increases at a rate of ( 135 % ) per hour. the number of cells increases by 135 per hour. the initial number of cells in the culture is 2,160. the initial number of cells in the culture is 1,600.

Explanation:

Step1: Analyze the exponential function form

The general form of an exponential function is \(y = a(1 + r)^x\), where \(a\) is the initial value and \(r\) is the growth rate. In the function \(f(x)=1600(1.35)^x\), when \(x = 0\), \(f(0)=1600(1.35)^0\).
Since any non - zero number to the power of \(0\) is \(1\) (\(a^0=1,a
eq0\)), we have \(f(0)=1600\times1 = 1600\). So the initial number of cells (when \(x = 0\)) is \(1600\).

Step2: Analyze the growth rate

For the growth rate, if \(y=a(1 + r)^x\), then \(1 + r=1.35\), so \(r=0.35 = 35\%\) (the growth rate per hour). It is not \(135\%\) (because \(135\%=1.35\) is the base of the exponential function, not the growth rate in percentage terms. The growth rate formula is \(r=(1.35 - 1)\times100\%=35\%\)). Also, it is not a linear increase (so not increasing by a constant number like \(135\) per hour as in a linear function \(y=mx + b\)).

Answer:

D. The initial number of cells in the culture is 1,600.